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A145236
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a(n) is the least positive integer such that if p_n is the n-th prime then (ceil(sqrt(a(n)p_n)))^2-a(n)p_n is a perfect square
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5
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2, 1, 1, 3, 5, 5, 9, 9, 13, 17, 19, 23, 25, 27, 31, 35, 41, 41, 47, 51, 51, 57, 61, 65, 73, 75, 77, 81, 83, 85, 99, 101, 107, 109, 117, 119, 125, 129, 133, 139, 145, 145, 155, 157, 161, 163, 173, 183, 187, 189, 193, 199, 201, 209, 215, 221, 225, 227, 233, 237, 239, 247
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Conjectures: 1) for n>=2 the sequence does not decrease; 2) for n>1 a(n) is odd; 3) a(n) could be equal to a(n+1) only for twins: p_(n+1)-p_n=2 (although there exist also twins for which a(n)<a(n+1)).
All these conjectures are proved using the formula a(n)= p_n- 2floor(sqrt(2p_n))+2, n>1. See also A145701 and A145714. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Oct 18 2008]
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MAPLE
| Contribution from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 02 2010: (Start)
A145236 := proc(n) local p, k, a ; p := ithprime(n) ; for k from 1 do ceil(sqrt(ceil(k*p))) ; a := %^2-k*p ; if issqr(a) then return k ; end if; end do: end proc:
for n from 1 do printf("%d, \n", A145236(n)) ; end do: (End)
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CROSSREFS
| Cf. A145016 A145022 A145023 A145047 A145048 A145049 A145050 A145215
Sequence in context: A187065 A187066 A174620 * A162206 A075248 A128325
Adjacent sequences: A145233 A145234 A145235 * A145237 A145238 A145239
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KEYWORD
| nonn
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AUTHOR
| Vladimir Shevelev (shevelev(AT)bgu.ac.il), Oct 05 2008, Oct 07 2008
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EXTENSIONS
| a(12)=23 (not 21). - Vladimir Shevelev (shevelev(AT)bgu.ac.il), Oct 16 2008
Extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 02 2010
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